Not at all. Take $f(x)=x^2$ on the unit interval. For examples of uniquely ergodic homeomorphisms of the Cantor set see [The prevalence of uniquely ergodic systems][1] by Jewett. A simple explicit example is provided by the boundary action of any hyperbolic automorphism of a homogeneous tree (attach a binary tree to each integer and look at the action of the integer shift on the space of ends of the whole construction). 


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=252604